Conditional Expectations for Unbounded Operator Algebras

Two conditional expectations in unbounded operator algebras (O∗-algebras) are discussed. One is a vector conditional expectation defined by a linear map of an O∗-algebra into the Hilbert space on which the O∗-algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an unbounded conditional expectation which is a positive linear map of an O∗-algebra onto a given O∗-subalgebra of . Here the domain D( ) of does not equal to in general, and so such a conditional expectation is called unbounded.